Define the partial order $>$: ${\bf u} >_{lex} {\bf v}$ if $u_i > v_$ for all $i$.
Define ${ >_{lex}}$ on $\N^n$ by using lexicographical ordering:
${\bf u} >_{lex} {\bf v}$ if the leftmost entry of ${\bf u} - {\bf v}$ is positive.
Ex: $(2, 3, 1) >_{lex} (2, 1, 3)$ since $(2, 3, 1) - (2, 1, 3) = (0, 2, -2)$
and the leftmost term $2 > 0$.
${\bf x}^{\bf u} = x_1^{u_1} x_2^{u_2} \cdot \cdot \cdot x_n^{u_n}$ is a monomial
Let $k$ be a field. Then if $f \in k[x_1, ..., x_n]$, $f = \S c_i x^{\bf u_i}$ for some $c_i
\in k$.
If the monomials are ordered such that ${\bf u_i} > {\bf u_j}$if $i < j$, then
$LC(f) = $ leading coefficient of $f = c_1$
$LM(f) = $ leading monomial of $f = x^{\bf u_i}$
$LT(f) = $ leading coefficient of $f = c_1 x^{\bf u_i}$